4/5/2021

Background

  • Screening
    • Pros: early detection

Background (cont’d)

Active surveillance programs

Prostate-Specific Antigen (PSA)

  • The blood level of PSA is often elevated in men with prostate cancer.

Active surveillance programs

Biopsy

  • Gleason score

    • 3 + 3 (6): low risk

    • ≥ 3 + 4/4 + 3 (≥7): intermediate risk

  • Drawback

    • Painful

    • Infections, sepsis

Research questions

  1. How can we make biopsy schedules in AS on an individual level?

  2. How can our personalized schedules relieve the burden of biopsies in AS at acceptable cost?

Previous work

Joint model

  • Measurement error

Previous work

Personalized schedules

Tomer, A., Nieboer, D., Roobol, M. J., Steyerberg, E. W., & Rizopoulos, D. (2022). Shared decision making of burdensome surveillance tests using personalized schedules and their burden and benefit. Statistics in medicine, 10.1002/sim.9347.

Data

Canary PASS Data

  • Target population: Patients diagnosed as Prostate cancer with Gleason score of 3+3

  • Sample size: 850 subjects recorded in the dataset, among which 833 participants had adequate PSA measurements for analysis.

Data features

  • Data types:

    • Repeated measurements: PSA values over time

    • Time-to-event data: time until progression

    • baseline variables: age at diagnosis and PSA density

  • Competing risk: treatment before progression (87/833)

  • Interval censoring: periodical examinations

Cause-specific Interval-censored Joint Model

  • Longitudinal submodel:

Cause-specific Interval-censored Joint Model

  • Longitudinal submodel:

\[ \begin{align} \log_2(\text{PSA}_i + 1)(t) &= \color{blue}{m_i(t)} + \epsilon_i(t)\\ \color{blue}{m_i(t)} &= \beta_0 + b_0 + \underbrace{(\beta_1 + b_1)\mathcal{C}^{(1)}_i(t) + (\beta_2 + b_2)\mathcal{C}^{(2)}_i(t) + (\beta_3 + b_3)\mathcal{C}^{(3)}_i(t)}_{\text{natural cubic splines for time variable}} + \underbrace{\beta_4(\text{Age}_i - 62)}_{\text{baseline covariate}} \end{align} \] where \(\mathcal{C}(t)\)is the design matrix for the natural cubic splines for time \(t\); the error term \(\epsilon_i(t) \sim t_3\)

Cause-specific Interval-censored Joint Model

  • Survival submodel:

The hazard, \(h_i^{(k)}(t)\), of experiencing event \(k\) (\(k = 1\) indicates progression; \(k = 2\) indicates treatment) for subject \(i\) at time \(t\) is: \[ \begin{align*} h_i^{(k)}(t \mid \boldsymbol{\mathcal{M}}_i(t)) = h_0^{(k)}(t)\exp\Big\{\gamma_k\text{density}_i + \alpha_{1k} \color{blue}{m_i(t)} + \alpha_{2k}\big[\color{blue}{m_i(t)} - \color{blue}{m_i(t-1)}\big]\Big\} \end{align*} \] where:

  • \(\boldsymbol{\mathcal{M}}_i(t)\) is the estimated true history of PSA levels for subject \(i\)

  • \(\color{blue}{m_i(t)}\) is the estimate of the true PSA level for subject \(i\) at \(t\)

  • \(\color{blue}{m_i(t)} - \color{blue}{m_i(t-1)}\) is the estimate of the PSA level change in the previous year from \(t\) for subject \(i\)

Cause-specific Interval-censored Joint Model

  • Visualization of \(\color{blue}{m_i(t)}\)

Cause-specific Interval-censored Joint Model

  • Visualization of \(\color{blue}{m_i(t)} - \color{blue}{m_i(t-1)}\)

Results

Joint model

Results

Trajectory of PSA levels

Results

Effect from yearly PSA change

Dynamic Prediction

More interest goes to the probability of getting progression, which can be achieved by cause-specific cumulative incidence function

\[\text{CIF}_j^{k=1}(t) = \int^t_0 \underbrace{h_j^{(1)}(s)}_{\text{instantaneous risk}\\ \text{of progression}}\underbrace{\exp\left\{-\sum^K\int^s_0h_j^{(k)}(\nu)d\nu\right\}}_{\text{overall survival function}}\]

Personalized schedules - Decision making

Whole personalized schedules

Whole personalized schedules (cont’d)

Whole personalized schedules (cont’d)

Whole personalized schedules (cont’d)

Choice of threshold (\(\kappa\))

  • Two indicators

    • Number of biopsies to be conducted

    • Detection delay

  • Higher threshold \(\rightarrow\) fewer biopsies but longer detection delay
  • Lower threshold \(\rightarrow\) more biopsies but shorter detection delay

Loss function

Simulation

Setting

  • 100 Training sets, 300 subjects
  • 100 Test sets, 200 subjects
    • 100 with progression
    • 100 without progression
  • Fit the joint models on the training set and create personalized schedules for the test set
  • Compared with the fixed schedules based on:
    • Number of biopsies
    • Detection delay
  • Fixed schedules include:
    • Annual biopsies
    • PASS protocol: at year 0.5, 1, 2 and afterwards biennially

Simulation

Results (Patients with progression)

Simulation

Results (Patients without progression)

Discussion

  • Impact of detection delay on the patients?

    • link with MISCAN
  • Progression-specific risk

  • Assume precision of biopsies

Thank you!

Supplementary

Number of biopsies \(\mathit{Nb}\)

  • The horizon of interest: \(t_{\text{hor}}\)
  • Generated schedules: \(S_i^\kappa = \{s_1, ..., s_{N_i}\}\)
  • Progression is not observed when generating the schedules
  • Expected number of biopsies are calculated
    • List all the possibilities of occurrence of progression

\[ \begin{align*} \mathit{Nb}_i(S_i^{\kappa}) = \begin{cases} 1, &t < T^*_i \leq s_1, k=1 \\ 2, &s_1 < T^*_j \leq s_2, k =1 \\ ... \\ N_i, &s_{N_i - 1} < T^*_i \leq s_{N_i}, k=1 \end{cases} \end{align*} \]

Number of biopsies \(\mathit{Nb}\) (cont’d)

Generally, the expected number of biopsies:

\[ \begin{align*} E\{\mathit{Nb}_i(S_i^\kappa)\} = \sum_{n=1}^{N_i} n \times \underbrace{\Pr\{s_{n-1}<T^*_i \leq s_n | T^*_i \leq s_{N_i}, k =1 \}}_{\text{probability of progressing between two biopsies}}, \ \ s_0 = t \end{align*} \]

Detection delay \(\mathit{Dd}\)

List all the possibilities of occurrence of progression:

\[ \begin{align*} \mathit{Dd}_j(S_j^{\kappa}) = \begin{cases} s_1 - T^*_i, &t < T^*_i \leq s_1, k=1 \\ s_2 - T^*_i, &s_1 < T^*_i \leq s_2, k =1 \\ ... \\ s_{N_i} - T^*_i, &s_{N_i - 1} < T^*_i \leq s_{N_i}, k =1 \end{cases} \end{align*} \]

Detection delay \(\mathit{Dd}\) (cont’d)

Generally, the expected detection delay is:

\[ \begin{align*} E\{\mathit{Dd}_j(S_i^\kappa)\} = \sum_{n=1}^{N_i} \big[s_n - \underbrace{E(T^*_i | s_{n-1}, s_{n}, v, k=1)}_{\text{expected progression time}}\big] \times \underbrace{\Pr\{s_{n-1}<T^*_i \leq s_n | T^*_i \leq s_{N_i}, k =1 \}}_{\text{probability of progressing between two biopsies}} \end{align*} \]